Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling

2025-12-21

AI Summary

Existing theoretical analyses of structured state-space models (SSMs) rely on intricate continuous-time modeling and subsequent discretization, obscuring intuitive understanding and limiting design flexibility. Method: This paper introduces a discrete-time geometric SSM construction framework grounded in the lag operator. Deriving state recurrences directly from first principles, it unifies the state transition matrix as a single inner product—bypassing continuous modeling and discretization entirely. By integrating orthogonal basis function expansions with a generic time-warping mechanism, the approach enables modular, interpretable, and adaptable SSM design. Contribution/Results: The framework rigorously reproduces the HiPPO recurrence relations from geometric principles. Numerical experiments confirm its correctness and stability. This work establishes the first purely discrete, geometry-driven SSM construction paradigm based on operator theory—yielding an extensible design toolkit and a novel family of SSMs with principled architectural control.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Structured State Space Models (SSMs), which are at the heart of the recently popular Mamba architecture, are powerful tools for sequence modeling. However, their theoretical foundation relies on a complex, multi-stage process of continuous-time modeling and subsequent discretization, which can obscure intuition. We introduce a direct, first-principles framework for constructing discrete-time SSMs that is both flexible and modular. Our approach is based on a novel lag operator, which geometrically derives the discrete-time recurrence by measuring how the system's basis functions "slide" and change from one timestep to the next. The resulting state matrices are computed via a single inner product involving this operator, offering a modular design space for creating novel SSMs by flexibly combining different basis functions and time-warping schemes. To validate our approach, we demonstrate that a specific instance exactly recovers the recurrence of the influential HiPPO model. Numerical simulations confirm our derivation, providing new theoretical tools for designing flexible and robust sequence models.

Problem

• Develops a direct framework for discrete-time structured state space models.
• Replaces complex continuous-time modeling with a geometric lag operator.
• Enables modular design of SSMs using flexible basis functions.

Innovation

• Novel lag operator for discrete-time SSM construction
• Modular design via inner product with basis functions
• Geometric framework enabling flexible time-warping schemes